<header>
    中值定理
</header>
<p>
    <span class="title">
        定理（罗尔中值定理）
    </span>
    若函数ƒ满足如下条件：
</p>
<ol>
    <li>
        ƒ在闭区间[a,b]上连续
    </li>
    <li>
        ƒ在开区间(a,b)上可导
    </li>
    <li>
        ƒ(a)=ƒ(b)
    </li>
</ol>
<p>
    则在(a,b)上至少存在一点ξ，使得
    <span class="oneline">
        ƒ<sup>'</sup>(ξ)=0
    </span>
</p>
<p>
    <span class="title">
        定理（拉格朗日中值定理）
    </span>
    若函数ƒ满足如下条件：
</p>
<ol>
    <li>
        ƒ在闭区间[a,b]上连续
    </li>
    <li>
        ƒ在开区间(a,b)上可导
    </li>
</ol>
<p>
    则在(a,b)上至少存在一点ξ，使得
    <span class="oneline">
        ƒ<sup>'</sup>(ξ)=<code>["division","ƒ(b)-ƒ(a)","b-a"]</code>
    </span>
</p>
<p>
    <span class="title">
        定理（柯西中值定理）
    </span>
    设函数ƒ和g满足
</p>
<ol>
    <li>
        在[a,b]上都连续
    </li>
    <li>
        在(a,b)上都可导
    </li>
    <li>
        ƒ<sup>'</sup>(x)和g<sup>'</sup>(x)不同时为零
    </li>
    <li>
        g(a)≠g(b)
    </li>
</ol>
<p>
    则在(a,b)上至少存在一点ξ，使得
    <code>
        ["division",["join",["rightTop","ƒ","'"],"(ξ)"],["join",["rightTop","g","'"],"(ξ)"]]
    </code>
    =
    <code>
        ["division","ƒ(b)-ƒ(a)","g(b)-g(a)"]
    </code>
</p>
<h2>
    洛必达法则
</h2>
<p>
    基于柯西中值定理，我们可以建立洛必达法则来帮助求解极限。
</p>
<h3>
    <code>["division","0","0"]</code>
    型不定式极限
</h3>
<p>
    <span class="title">
        定理
    </span>
    若函数ƒ和g满足：
</p>
<ol>
    <li>
        <code>["limt",["join","x→",["rightBottom","x","0"]],"ƒ(x)"]</code>=
        <code>["limt",["join","x→",["rightBottom","x","0"]],"g(x)"]</code>=0
    </li>
    <li>
        在点x<sub>0</sub>的某个空心领域U<sup>0</sup>(x<sub>0</sub>)上两者都可导，且g<sup>'</sup>≠0
    </li>
    <li>
        <code>["limt",["join","x→",["rightBottom","x","0"]],["division",["join",["rightTop","ƒ","'"],"(x)"],["join",["rightTop","g","'"],"(x)"]]]</code>=A
        (A可为实数，也可为±∞或∞)
    </li>
</ol>
<p>
    则
    <span class="oneline">
        <code>["limt",["join","x→",["rightBottom","x","0"]],["division","ƒ(x)","g(x)"]]</code>=
        <code>["limt",["join","x→",["rightBottom","x","0"]],["division",["join",["rightTop","ƒ","'"],"(x)"],["join",["rightTop","g","'"],"(x)"]]]</code>=A
    </span>
</p>
<h3>
    <code>["division","•","∞"]</code>
    型不定式极限
</h3>
<p>
    <span class="title">
        定理
    </span>
    若函数ƒ和g满足：
</p>
<ol>
    <li>
        在x<sub>0</sub>的某个右领域U<sup>0</sup><sub>+</sub>(x<sub>0</sub>)上两者可导，且g<sup>'</sup>(x)≠0
    </li>
    <li>
        <code>
            ["limt",["join","x→",["rightTop",["rightBottom","x","0"],"+"]],"g(x)"]
        </code>
        =∞
    </li>
    <li>
        <code>
            ["limt",["join","x→",["rightTop",["rightBottom","x","0"],"+"]],["division",["join",["rightTop","ƒ","'"],"(x)"],["join",["rightTop","g","'"],"(x)"]]]
        </code>
        =A (A可为实数，也可为±∞或∞)
    </li>
</ol>
<p>
    则
    <span class="oneline">
        <code>
            ["limt",["join","x→",["rightTop",["rightBottom","x","0"],"+"]],["division","ƒ(x)","g(x)"]]
        </code>
        =A
    </span>
</p>